Projection of matrix onto subspace

Question

I am confused about why the orthogonal projection of matrices onto subspaces is given by a change-of-basis-like formula. For example, in the below image from these notes, why is the orthogonal projection of matrix A onto the subspace $V_m$ given by $V_m^* A V_m$?

When I search online, I come across a formula like (for example, from the Wikipedia article), which does not look at all like $V_m^* A V_m$.

Answer 1

I have the same question, but don't have the reputation to comment.

It's worth noting that you have two different $A$ matrices in your question - the $A$ in the standard projection formula corresponds to your $V_m$. Because the column-vectors of the subspace are orthonormal, $V_m^T V_m = I$, and so the projection matrix (in this notation) is $P \equiv V_m V_m^T$. Here is where I get stuck.